menthol has this cooling effect, something like a very mild anesthetia that distracts you from body pains. the reason people like inhaling menthol is beause of this relaxing effect, so say your nose is congested, there's this sort of decongesting feeling because the scent from menthol makes you feel calmer and more relaxed. people report feeling better even if objectively it doesn't significantly open up the airways; i personally have allergies and menthol inhalers dont totally cure my stuffy nose, but hey at least i feel a little better.
it's kinda the same reason why toothpaste is usually minty; the mint flavor doesnt really do much hygiene-wise, but because there's a lasting cooling effect, we feel clean for a while. menthol is just kinda there to make us feel good.
speaking of, as a filipina, i know so many people who like these kinda of symptomatic remedies. my friends and i have these vic's keychain menthol inhalers cuz we're always so stressed lmao. it became a trend at one point to have these pastel inhalers lol. my grandmother also had her own kind of oil remedy "white flower" which had a really strong smell.
here's what my lil keychain inhaler looks like lol. i have one but my bf gave me another jic i'd be too stressed that i needed another